The p-widths Database

For every closed, smooth, Riemannian manifold M of dimension at least 2, there is an associated non-decreasing sequence {ωp(M)}p=1,2,... of positive real numbers. These numbers are called the p-widths of M. Here is the heuristic definition: Using geometric measure theory, we consider a certain space 𝒵 of "all objects in M of codimension 1 without boundary". By Almgren [Alm62], 𝒵 is weakly homotopy equivalent to ℝℙ∞, so its cohomology ring

H*(𝒵; ℤ2) = ℤ2[λ].

Now, a map Φ: X → 𝒵, with X being any finite simplicial complex, is called a p-sweepout if Φ*(λp) ≠ 0. Denoting by 𝒫p the set of p-sweepouts, we define the p-width ωp(M) of M as

infΦ∈𝒫p supx∈dmn(Φ) area(Φ(x)),

where dmn(Φ) denotes the domain of Φ. For a precise definition of p-widths, see Marques-Neves [MN17].

Initiated by Marques-Neves and continued by many others, there has been a spectacular development in using min-max theory to construct minimal hypersurfaces. This includes, for example, Marques-Neves' proof [MN14] of the Willmore conjecture, and Song’s proof [Son18] of Yau’s conjecture on the existence of infinitely many immersed closed minimal surfaces in 3-manifolds [Yau82, p.689]. Notably, the p-widths play a crucial role in min-max theory.

In the following, we record results regarding explicit values of p-widths. Newly added results are in red. Comments are welcome!

Spheres:

Let Sn be the unit n-sphere.

• ω1(S2) = ω2(S2) = ω3(S2) = 2π
  ω4(S2) = ... = ω8(S2) = 4π [Aie19]

• ωp(S2) = 2π⌊√p⌋ for all p [CM23]

• ω1(S3) = ... = ω4(S3) = 4π

• ω5(S3) = ... = ω8(S3) = 2π2 [Nur16, Mar23]

• 2π2 < ω9(S3) < 8π [Nur16]

• ω13(S3) < 8π [CL23]

• ωp(Sn) = area(Sn-1) for n ≥ 2, p = 1 to n+1

Balls:

Let Bn be the unit n-ball.

• ω1(B2) = ω2(B2) = 2
  ω3(B2) = ω4(B2) = 4 [Don22]

• ω5(B2) > 4 [DM22]

• ω1(B3) = ω2(B3) = ω3(B3) = π

• ω6(B3) < 2π [Chu23]

• ωp(Bn) = area(Bn-1) for n ≥ 2, p = 1 to n

Projective spaces:

Let RPn be the real projective space Sn/ℤ2 and CPn the complex projective space S2n+1/U(1).

• ω1(RP3) = ω2(RP3) = ω3(RP3) = π2 

ω5(RP3) > π2 

ω9(RP3) ≤ 4π [BL22]

• ω1(RP4) = ω2(RP4) = 8/3 · 1/√3 · π2
  ω1(RP5) = ω2(RP5) = 2π2
  ω1(RP6) = ω2(RP6) = 24/25 · √(3/5) · π3
  ω1(RP7) = ω2(RP7) = π4/4 [Ram19, BL23b]

• ω1(CP2) ≤ 3/8 · √3 · π2
  ω1(CP3) ≤ π3/4 [Ram19]

Len spaces: 

Let L(p, q), with p > q, be the lens space S3/ℤp.

• ω1(L(p, q)) = 2π2/p [BL23a]

Bibliography:

• [Aie19] Aiex, Nicolau Sarquis. "The width of ellipsoids." Communications in Analysis and Geometry 27.2 (2019): 251-285.

• [Alm62] Almgren Jr, Frederick Justin. "The homotopy groups of the integral cycle groups." Topology 1.4 (1962): 257-299.

• [BL22] Batista, Márcio, and Anderson Lima. "Min-max widths of the real projective 3-space." Transactions of the American Mathematical Society 375.07 (2022): 5239-5258. 

• [BL23a] Batista, Márcio, and Anderson Lima. "A short note about 1-width of Lens spaces." Bulletin of Mathematical Sciences 13.01 (2023): 2250005. 

• [BL23b] Batista, Márcio, and Anderson De Lima. "The first and second widths of the real projective space." Proceedings of the American Mathematical Society (2023). 

• [CL23] Chu, Adrian Chun-Pong, and Yangyang Li. "A strong multiplicity one theorem in min-max theory." arXiv preprint arXiv:2309.07741 (2023).

• [CM23] Chodosh, Otis, and Christos Mantoulidis. "The p-widths of a surface." Publications mathématiques de l'IHÉS 137.1 (2023): 245-342. 

• [Chu23] Chu, Adrian Chun-Pong. "A free boundary minimal surface via a 6-sweepout." The Journal of Geometric Analysis 33.7 (2023): 230. 

• [Don22] Donato, Sidney. "The first p-widths of the unit disk." The Journal of Geometric Analysis 32.6 (2022): 177. 

• [DM22] Donato, Sidney, and Rafael Montezuma. "The first width of non-negatively curved surfaces with convex boundary." arXiv preprint arXiv:2211.14815 (2022).

• [MN14] Marques, Fernando C., and André Neves. "Min-max theory and the Willmore conjecture." Annals of Mathematics (2014): 683-782.

• [MN17] Marques, Fernando C., and André Neves. "Morse index and multiplicity of min-max minimal hypersurfaces." Cambridge Journal of Mathematics 4.4 (2016): 463-511.

• [Mar23] Marques, Fernando C., personal communication.

• [Nur16] Nurser, Charles Arthur George. Low min-max widths of the round three-sphere. Diss. Imperial College London, 2016. 

• [Ram19] Ramírez-Luna, Alejandra. "Compact minimal hypersurfaces of index one and the width of real projective spaces." arXiv preprint arXiv:1902.08221 (2019).

• [Son18] Song, Antoine. "Existence of infinitely many minimal hypersurfaces in closed manifolds." arXiv preprint arXiv:1806.08816 (2018).

• [Yau82] Shing-Tung Yau. Seminar in Differential Geometry, Problem Section. pages 669-706, Annals of Mathematics Studies (1982): 102, Princeton University Press, Princeton, N.J.